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Symmetry of molecular Rydberg states revealed by XUV transient
absorption spectroscopy
Peng, Peng; Marceau, Claude; Hervé, Marius; Corkum, P. B.; Naumov, A.
Yu.; Villeneuve, D. M.
Symmetry of molecular Rydberg states revealed
by XUV transient absorption spectroscopy
Peng Peng
1*, Claude Marceau
1, Marius Hervé
1, P.B. Corkum
1, A.Yu. Naumov
1& D.M. Villeneuve
1*
Transient absorption spectroscopy is utilized extensively for measurements of bound- and quasibound-state dynamics of atoms and molecules. The extension of this technique into the extreme ultraviolet (XUV) region with attosecond pulses has the potential to attain unprecedented time resolution. Here we apply this technique to aligned-in-space molecules. The XUV pulses are much shorter than the time during which the molecules remain aligned, typically <100 fs. However, transient absorption is not an instantaneous probe, because long-lived coherences re-emit for picoseconds to nanoseconds. Due to dephasing of the rotational wavepacket, it is not clear if these coherences will be evident in the absorption spectrum, and whether the properties of the initial excitations will be preserved. We studied Rydberg states of N2and O2from 12 to 23 eV. We were able to determine the polarization direction of the electronic transitions, and hence identify the symmetry of the final states.
https://doi.org/10.1038/s41467-019-13251-w OPEN
1Joint Attosecond Science Laboratory, National Research Council and University of Ottawa, Ottawa, ON K1A 0R6, Canada. *email:pengpengnrc@gmail.com; david.villeneuve@uottawa.ca
123456789
X
UV transient absorption spectroscopy uses coherent extreme ultraviolet (XUV) pulses to probe the electronic structure of atoms and molecules. Attosecond transient absorption spectroscopy (ATAS) uses high harmonic generation to create a broad spectrum in the XUV or soft X-ray range. The broad spectrum permits the recording of many absorption lines in parallel1–17. ATAS can probe valence electrons throughexcitation to high-lying electronic states1–14. ATAS can also excite core-level electrons to reveal molecular structure using techniques such as near-edge X-ray absorption fine structure (NEXAFS)15–18, for example around the carbon K-edge at 284 eV. Pump-probe type experiments in which two-photon processes are driven by a combination of an attosecond pulse and an infrared pulse have revealed coupling between states and the production of light-induced states4,7. Shifts in absorption features have been used to observe vibrational motion in bromine molecules14.
When linear molecules are irradiated by a infrared pump pulse whose duration is much shorter than the rotational period, nonadiabatic field-free alignment is achieved19–21. Each initial
thermally-populated rotational eigenstate Jj0; M0i will expand to
a rotational wave packet. ΨJ 0;M0ðtÞ ¼ X J;M aJ0;M0 J;M ðtÞ J; Mj i; ð1Þ
where J is the rotational quantum number, M is the projection on the polarization axis and aJ0;M0
J;M ðtÞ is the amplitude for each J; Mj i
state. The time evolution of the wave packet can be calculated by solving the time-dependent Schrodinger equation. The degree of alignment is characterized by:
hcos2θiJ
0;M0ðtÞ ¼ hΨJ0;M0ðtÞjcos
2θjΨ
J0;M0ðtÞi; ð2Þ in which θ is the angle of the molecular axis to the polarization axis. Finally, thermally averaged values of the degree of alignment are obtained as:
hcos2θiðtÞ ¼X
J0;M0
ρJ0;M0ðTÞhcos
2θi
J0;M0ðtÞ; ð3Þ
where T is the rotational temperature and ρJ
0;M0ðTÞ is the Boltzmann weight function for the jJ0; M0i initial states.
hcos2θiðtÞ varies with time after the aligning laser pulse as various
rotational states go in and out of phase. The probe attosecond pulses arrive at a revival of the rotational wavepacket at which the value of hcos2θi maximizes. The electronic excitation is in the molecular frame, but observations occur in the laboratory frame; the frames are related through the Wigner rotation matrix. The free induction decay resulting from the induced polarization in the molecule frame continues to radiate while the rotational wavepacket dephases. At times the radiation associated with different rotational eigenstates will be out of phase and will lead to cancellation of the emission. In addition, the rotational con-stants of the lower and upper states will be different. Therefore it needs to be demonstrated that the time-integrated absorption of aligned molecules will exhibit the characteristics of the attosecond excitation.
In the following, we use XUV transient absorption spectro-scopy to record the molecular absorption spectra of N2and O2. The short pulse duration of attosecond pulse trains (typically 5–30 fs, but as short as 50 as refs. 22,23) permits using a
near-infrared (NIR) laser pulse to impulsively align the molecular sample19–21, allowing the absorption measurements to be made
in a sample in which the molecular axes can be aligned relative to the polarization axis of the probing XUV field. This is not pos-sible in synchrotron experiments due to their relatively long pulse durations. We will show that we can discriminate between par-allel and perpendicular transitions for linear molecules, and hence
identify the symmetry of the final states. In the case of O2, the
observed symmetry of some states differs from previous assign-ments. This same approach can be applied to larger molecules, for example asymmetric top molecules24.
Results
XUV transient absorption in N2 and O2. Figure 1a illustrates the experimental arrangement. A 10 fs NIR pulse centered at 800 nm was focused into a pulsed gas jet of xenon for high harmonic generation (see Methods for more details). The band-width of each harmonic was 2.3 eV and the total spectrum spanned 12–24 eV, as shown by the black curve in Fig.1b. A 70 fs NIR pulse was used to perform impulsive alignment on the sample molecules. It was combined with the XUV pulse with a holey mirror. The alignment pulse was displaced from the center of the holey mirror while the XUV pulse passed through the hole. Both the XUV and the alignment pulses were vertically polarized, they propagated noncollinearly and intersected at the absorption gas jet with a crossing angle of 18 mrad. The transmitted XUV spectra were recorded at different time delays τ between the alignment and XUV pulses. The blue (red) curve in Fig. 1(b) shows the transmitted XUV spectrum at aligned delay τ ¼ 8:57 ps (anti-aligned delay τ ¼ 8:32 ps) for N2. About 10,000
laser shots were accumulated at each delay point.
We define the differential absorption spectrum to be the differential optical density ΔODðE; τÞ ¼ log10½IonðE; τÞ=
IoffðE; τÞ. Here, IonðE; τÞ and IoffðE; τÞ represent the XUV spectra after transmission through the sample, taken with and without the alignment field, respectively. This is effectively the XUV absorption of the aligned medium minus that of the randomly aligned medium Fig. 2a shows the static absorption of isotropic N2 measured by
blocking the NIR alignment pulse. The ground state configuration may be written as ð1σgÞ2ð1σuÞ2ð2σgÞ2ð2σuÞ2ð1πuÞ4ð3σgÞ2X1Σþg.
The adiabatic ionization energies corresponding to different final states of Nþ2 are ð3σgÞ 1X2 Σþ g ¼ 15:58 eV, ð1πuÞ 1 A2Π u¼ 16:70 eV, ð2σuÞ 1B2Σþ
u ¼ 18:75 eV25–27, as shown by the thick
gray lines in Fig.2a. The XUV pulse excites numerous overlapping vibrational series associated with states of Rydberg and valence character. The lifetimes of these states are quite different. The
a
b
Xenon gas jet
80 µm Quartz Filter
10 fs NIR HHG Toroid
Holey mirror 70 fs NIR
Absorption gas jet
Grating 1 0.8 0.6 0.4 0.2 0 Intensity (arb .u.) 13 14 15 16 17 18 19 20 21 22 23 Energy (eV) Without N2 Aligned N2 Antialigned N2
Fig. 1Experimental setup and XUV spectra. a Schematic of the
experimental setup. A broad XUV spectrum is generated by focusing a 10 fs NIR pulse into a xenon gas jet to generate an attosecond pulse train. The XUV is refocused in a second gas jet containing the target gas. A longer NIR pulse is used to impulsively align the molecules in the second gas jet. bXUV spectra taken without the target gas (black curve), after absorption by N2at the delay times corresponding to alignment (blue curve) and anti-alignment (red curve).
states between 17 and 18.5 eV have short lifetimes of 10 fs due to autoionization12 and the absorption peaks show wide
asymmetric Fano line shapes, whereas states below the ionization potential (15.58 eV) have long lifetime >ps28,29 and the
absorption peaks show narrow symmetric Lorentzian line shapes. Following the assignment of30–33, the absorption features are
summarized in Table 1. We use red (blue) lines to label the excited states with1Σþ
u (1Πu) symmetry. Due to the finite energy
resolution of the XUV spectrometer (70 meV), the full structure of the excited states cannot be resolved. Figure 2b shows the differential absorption spectrum measured around the full rotational revival delay of N2. The alignment-induced absorbance modulation can be seen across the whole energy range, which shows that both long-lived states and short-lived autoionizing states are sensitive to molecular alignment at the time of the
probe signal. Figure2d shows lineouts versus time delay for the absorption peaks at 15.28 eV (marked by the blue arrow) and 17.14 eV (marked by the red arrow); the different modulation depth and polarity are evident.
Figure2b clearly shows that there are two types of absorption features: those that maximize when the molecular axis is parallel to the XUV polarization direction (τ ¼ 8:57 ps), and those that maximize when the molecular axis is perpendicular (τ ¼ 8:32 ps). For example, at 15 eV, the direction changes, as it does at 18.2 eV. Visually one can identify regions of parallel or perpendicular electronic transitions, illustrating the power of ATAS with aligned molecules.
We now turn to O2. Figure3a shows the static absorption of
isotropic O2. The ground state may be written as
ð1σgÞ2ð1σuÞ2ð2σgÞ2ð2σuÞ2ð3σgÞ2ð1πuÞ4ð1πgÞ2X3Σg. The adia-batic ionization energies corresponding to different final states of Oþ2 are ð1πgÞ 1X2Π
g¼ 12:07 eV, ð1πuÞ 1
a4Π u¼
16:1 eV, ð1πuÞ 1A2Πu¼ 17:05 eV, ð3σgÞ 1b4Σg ¼ 18:17 eV,
ð3σgÞ 1B2Σ
g ¼ 20:3 eV, ð2σuÞ 1c4Σu ¼ 24:56 eV34,37, as
shown by the thick gray lines in Fig. 3(a). Following the assignment of34–36, the absorption features are summarized in Table2. Similarly, we use red (blue) lines to label the excited states with 3Σ
u (3Πu) symmetry. Figure 3b shows the
differential absorption spectra and Fig. 3d shows lineouts for the absorption peak at 13.43 eV (marked by blue arrow) and 15.18 eV (marked by red arrow).
The experimental results in Figs. 2 and 3 show that the transient absorption signal indeed depends on the orientation of the molecules at the time of the XUV pulse, answering the question that we posed in the introduction. But it is not clear why this is true. The pump pulse creates a coherence between rotational states in the electronic ground state. At a later time the XUV pulse coherently excites the ensemble of molecules to a number of excited electronic states. The superposition of states results in an induced time-dependent polarization which will radiate an electric field for many picoseconds. Because the emission is out of phase with the XUV pulse, the time-integrated emission will appear as absorption5,9. For an atom, the emission will decay exponentially over picoseconds or nanoseconds due to the lifetime of the excited state, and thereby determines the
Lifetime > ps Lifetime ~10 fs 13 14 15 16 17 18 19 20 21 22 23 Δ OD 0.05 0 –0.05 9 8.5 8 9 8.5 8 Dela y (ps) Dela y (ps) 0.1 0 –0.1 13 14 15 16 17 18 19 20 21 22 23 Energy (eV) < cos2> –1/3 X2Σ+ g A2Πu B 2 Σ+u OD Δ OD Experiment Δ OD Calculation 0.8 0.6 0.4 0.2 OD 0.1 0.05 0.15 0.15 0 –0.05 0.1 0.05 0 –0.05 a d e c b
Fig. 2Experimental and calculated XUV transient absorption in aligned N2. a Static absorption of isotropic N2molecules from 12.7 to 23.7 eV. The positions
of vibrational levels of the valence b1Π
uand b’1Σþu states and of the Rydberg series built on X2Σ þ g; A
2Π
uand B2Σþu N þ
2 cores are indicated, parallel and
perpendicular transitions are labeled by red and blue lines respectively. b Experimental and c calculated differential absorption spectra around the full revival of aligned N2molecules. d Modulations of the spectrum at 15.28 eV (blue circle) and 17.14 eV (red circle). e Calculated alignment degree hcos2θi 1=3.
Table 1 Assignment ofN2absorption spectrum.
ϵ(eV) Transition Description Nþ
2core 12.7–15.58 ð1πuÞð3σgÞ ! ð1πgÞ2 b1Πau Valence 2σu! 1πg 3σg! 3σu b’1Σþ a u Valence 1πu! 1πg 3σg! npσu CYb X2Σþg 3σg! npπu WJb 1πu! 3sσg oa3 A 2Π u 15.58–16.70 1πu! ndδg WTb 1πu! ndσg OTb Unassigned UNc;d 16.70–18.75 2σu! ndσg HAb;c B2Σþu 2σu! ndπg HEb;c 2σu! nsσg New OTb;c
The experimental absorption spectrum of N2, shown in Fig.2a, consists of numerous
overlapping vibrational series associated with states of Rydberg and valence character. The absorption features are summarized by the photon energy ϵ, the column labeled “Description" is the name of the electronic transitions in the “Transition" column, the “N2+core" column is the
ionic state, to which the Rydberg series converge
CYCarroll Yoshino, WJ Worley Jenkins, WT Worley third, OT Ogawa Tanaka, UN Unassigned, HAHopfield absorption, HE Hopfield emission
aReference30
bReference31
cReference32
spectral linewidth of the absorption feature. For molecules, the emitted polarization is more complicated, because the molecules continue to rotate after the XUV pulse.
Calculation of induced polarization. To understand the time-domain picture of re-emission, we illustrate the time-varying polarization for a simplified rigid rotor modeled on the N2
molecule. We consider rotational states with J in the range 0–10. We start with only J ¼ 0; M ¼ 0 and create a rotational wavepacket in the ground electronic state 0j i by coherently populating the even Js via Raman transitions. This determines the set of aJ. The rotational wavepacket has a full revival at
t ¼ 8:5 ps, just after an anti-revival at t ¼ 8:2 ps. At a delayed
time tx, the XUV pulse creates population in an excited elec-tronic state 1j i through dipole transitions. We consider a par-allel transition, i.e. one which is allowed when the XUV polarization is parallel to the molecular axis. Rotational selec-tion rules require that ΔJ ¼ ±1, so a set of odd-J rotaselec-tional states are populated with complex amplitudes bJ, creating a
second rotational wavepacket in the excited electronic state.
The induced polarization is proportional to the time-dependent dipole moment of the molecule.
dðtÞ ¼ hψ1ðtÞjrjψ0ðtÞi þ c:c: ¼ P JJ0 aJbJ0eiE 1 J0ðt txÞe iE0Jtd jjhJ0Mj cos θjJMi þ c:c:; ð4Þ here E0
J and E1J0 are the energies of the eigenstates, including both rotational and electronic parts, djj is the parallel compo-nent of the electronic transition dipole moment in the mole-cular frame. dðtÞ will have complex behavior in time, especially if the two electronic states have different rotational constants. See the Supplementary Note 1 for more details. Figure4shows a calculation of the induced polarization for this simplified molecule. The polarization results in free induction decay from the excited states. The re-emission is out of phase with the incident attosecond pulse, resulting in absorption lines in the transmitted spectrum5,9,10. A stronger polarization signal is
associated with stronger absorption. The high frequency oscil-lations are mainly determined by the electronic transition fre-quency (14 eV in this calculation, not visible on this time scale). The low frequency oscillations are induced by the coherence
X2Πg a4Π u A 2Π u b 4Σ– g B 2 Σ–g 13 14 15 16 17 18 19 20 21 22 Δ OD 0.03 0 –0.03 12 11.5 11 12 11.5 11 Dela y (ps) Dela y (ps) 13 14 15 16 17 18 19 20 21 22 0.1 0 –0.1
< cos2> –1/3 Energy (eV)
0.6 0.4 0.2 OD 0.1 0.05 0 –0.05 0.1 0.05 0 –0.05 Δ OD Experiment Δ OD Calculation OD a d e c b
Fig. 3Experimental and calculated XUV transient absorption in aligned O2. a Static absorption of isotropic O2molecules from 12 to 22.9 eV. The positions of vibrational levels of Rydberg series built on several Oþ2 cores are indicated, parallel and perpendicular transition are labeled by red and blue lines respectively. b Experimental and c calculated differential absorption spectra around the full revival of aligned O2molecules. d Modulations of the spectrum at 13.43 eV (blue circle) and 15.18 eV (red circle). e Calculated alignment degree <cos2θ> 1=3.
Table 2 Assignment ofO2 absorption spectrum.
ϵ(eV) Transition Description Oþ
2core 12.3–17.05 1πu! 3sσg Ha a4Πu 1πu! 4sσg Ib;c 1πu! 3dδg I’b;c 1πu! 3dσg I”b;c 1πu! 3sσg Ja A2Πu 17.05–20.3 3σg! npσu Stronga b4Σg 3σg! npπu Weaka 3σg! npσu Stronga B2Σg 3σg! npπu Weaka 20.3–21.5 2σu! 3sσg Stronga c4Σu
The experimental absorption spectrum of O2, shown in Fig.3a, consists of numerous
overlapping vibrational series associated with states of Rydberg and valence character. The absorption features are summarized by the photon energy ϵ, the column labeled “Description" is the name of the electronic transitions in the “Transition" column, the “O2+core" column is the
ionic state, to which the Rydberg series converge
aReference34 bReference35 cReference36 1 0.5 Alignment Anti-alignment Induced polarization 0 –0.5 –1 0 5 10 15
Time after XUV excitation (ps)
20 25
Fig. 4Calculated induced dipole moment for a rigid rotor linear molecule based on N2. A rotational wavepacket is created in the ground state by the pump pulse, and the XUV pulse arrives at a later time tx. If txis during a rotational revival (blue), then a greater polarization is induced. On the other hand, if txis when the molecules are perpendicular to the polarization, then
fewer molecules are excited. A larger induced polarization signal causes a greater absorption in the frequency spectrum. The rotational constants are B0¼ 2 cm 1and B1¼ 1:9 cm 1.
between different rotational states and different rotational rates between the ground and excited states. This dipole leads to an absorption structure around 14 eV, which consists of several peaks, the separations of which are related to the rotational energy (meV, which cannot be resolved by our XUV spec-trometer). Probe times txat the rotational revival (blue) and the anti-revival (red) are shown. The orientation-dependent beha-vior is evident during the first picosecond. For aligned mole-cules, the polarization signal is greatest immediately after excitation. For anti-aligned molecules, the signal is at first small, then increases as the molecules move into alignment 300 fs later. The overall signal is greater when the probe pulse arrives during the molecular alignment. This agrees with our experimental result: parallel transition shows stronger absorp-tion when the XUV pulse arrives during a time of maximum alignment to the polarization direction.
One can understand this behavior classically. If the probe pulse arrives during a rotational revival, then more excited state population is produced for a parallel transition, and the excited molecules will also be parallel to the polarization direction. But the molecules in the ground and excited states will rotate at different rates, meaning that the induced dipole will modulate at their difference frequency. This leads to the modulations seen in Fig.4. Frequency domain model. We now proceed to model the expected absorption in the frequency domain. More details are provided in the Supplementary Note 2. Impulsive alignment of a rigid rotor results in a distribution of molecular angles θ relative to the polarization axis ^z of the laser pulse. The distribution is cylindrically symmetric about ^z and varies in time based on the rotational constant of the molecule and its nuclear spin statis-tics19. The degree of alignment is characterized by the expectation value of the cosine-squared of the angle of the molecular axes to ^
z, namely hcos2θiðtÞ. For an isotropic sample, hcos2θi ¼ 1=3; for
a sample perfectly aligned along direction ^z it is 1. At char-acteristic times called rotational revivals, the molecular axis dis-tribution achieves its maximum degree of alignment. Just before a full revival, the molecular axes lie mostly in a plane perpendicular to ^z, call anti-alignment, quickly followed by a maximum degree of alignment parallel to ^z. Figures2e and3e show the calculated evolution of hcos2θiðtÞ 1=3 in the region of the rotational
revival.
For molecules of D1h symmetry group, all bound-bound
electronic transitions must be either parallel or perpendicular to the molecular axis38. The measured absorption will then be a
mixture of both types, weighted by the degree of alignment, σðE; tÞ ¼ hcos2θiðtÞ σjjðEÞ þ 1 hcos2θiðtÞ
σ?ðEÞ; ð5Þ
where σjjðEÞ and σ?ðEÞ are absorption cross sections for parallel and perpendicular transitions. The absorption cross section for bound-bound transitions are taken from the high-resolution spectra of ref. 25 for N
2 and ref. 34 for O2, including their
symmetry assignments (summarized in Tables 1 and 2). The absorption cross sections for bound-free transitions above each ionization threshold are assumed to be flat in energy and are fit to the experiment, as described in the Supplementary Note 2. The calculated transmitted spectra are convolved with the experi-mental instruexperi-mental response function of 70 meV width, and the predicted differential absorption spectrum ΔOD is calculated in the same way as for the experimental data. The model results are shown in Fig.2c for N2and Fig.3c for O2.
The experimental and the calculated results are in good agreement. The significant difference around 13 eV for N2may
come from several factors. First, the XUV signal itself is weak around 13 eV as shown in Fig.1(b). Second, the absorption cross
section is huge (400 Mb) as shown in the Supplementary Fig. 3, which means the XUV spectrum after absorption is very weak and the signal-to-noise ratio is poor in this energy range. Third, the second order diffraction of higher energy XUV photons (26 eV) falls onto 13 eV, which may pollute the signal. The simulated signal contrast is also larger than the measured contrast. One possible explanation is the assumption of full coherence of the rotational wavepacket in the ground state, by the use of Schrodinger’s equation. A more accurate simulation would instead use a density matrix approach in which a degree of incoherence in the rotational wavepacket is included. Another possible reason is the assumption of a constant spectrometer resolution in the whole spectrum range in the simulation, whereas the real resolution changes with photon energy (better resolution at lower photon energy).
Discussion
The photoabsorption spectra of small molecules have been extensively studied using high-resolution XUV monochromators at synchrotron radiation facilities25,34,39–42. Large numbers of
observed bands in the absorption spectra have been grouped into Rydberg series converging to different vibrational levels of various ion states. But one cannot get information on symmetries of photoexcited states from traditional photoabsorption spectro-scopy without the help of theoretical calculations43. The Rydberg orbitals are usually determined by calculating the effective quantum number n* and quantum defect δ, then the symmetry of the Rydberg state can be classified according to the build-up principles given by Herzberg44. Normally, assignments can be
confirmed from the rotational analysis of the Rydberg transition or from the analysis of fluorescence polarization. However, few such data are available. In O2, the lifetime broadening due to rapid autoionization and/or predissociation results in the absence of clearly resolved rotational structure34,45. So the symmetry assigned to some Rydberg series are tentative and various inter-pretations exist35,36,46–50.
Despite extensive experimental and theoretical work on the assignments of the Rydberg series converging onto the Nþ2 X, A, and B states, a consistent interpretation has yet to emerge46. The
main difference between these assignments concerns the Rydberg series converging onto the Nþ2 A threshold. In Fig.5(a), the black line is the absorption spectrum adapted from a synchrotron study25. Two series known as Worley’s third series and Ogawa-Tanaka series have been observed, labeled as WT and OT. However, some of the most prominent features in the absorption spectrum are not included in either of these two series and are labeled as UN. Figure5b is a magnified portion of Fig.2b from 15.72 to 16.76 eV. We can see most of the UN peaks are con-sistent with parallel transitions (negative for anti-alignment and positive for alignment).
For better comparison, we choose the alignment and anti-alignment delay points, labeled by the red and blue arrows in Fig.5b. We define an asymmetry signal: P ¼ ΔODðAlignmentÞ ΔODðAnti alignmentÞ, shown by the black line in Fig.5c. If we assume that the UN series have 1Σþ
u symmetry (parallel
transi-tions), the predicted spectrum is shown by the red line. On the other hand, assuming1Π
u symmetry (perpendicular transitions)
gives the blue line. The red line shows much better agreement with the experimental result, which suggests that these unas-signed peaks have1Σþu symmetry. Consider the Rydberg orbital to be nlλg=u, where n is the principal quantum number, l is the
orbital angular momentum quantum number, λ is the quantum number of the projection of orbital angular momentum onto the internuclear axis and g=u means symmetric/antisymmetric orbi-tal. Since the vibrational spacings in the UN series resemble those of the Nþ2A2Πu state32,33, these series should originate from the
promotion of a 1πu electron. So the Rydberg orbital symmetry
must be g, which means l is even44. Our experiment identified
that the UN series are parallel transitions, so λ should be π, which means λ ¼ 1. Since l λ, l must be 2, which means a d orbital.
We also calculated the quantum defects δ by νnðvÞ ¼
ν1ðvÞ R=ðn δÞ250, where νnðvÞ is the Rydberg state with
principal quantum number n and in the corresponding v vibra-tional level33, ν
1ðvÞ is the series limit51; R ¼ 109737:316 cm 1is
the Rydberg constant. For molecules built up from atoms of the first two rows of the periodic table, δ 1 for s orbital, δ 0:7 for
p orbital and δ 0 for d orbital52. The result δ 0:2 also
suggests a d orbital. So the transitions for these UN series are 1πu! ndπg. Usually quantum defect calculations are not enough
for the Rydberg orbital assignment, in this case, δ 0:2 sug-gests a d orbital, but it can be dσg, dπg, or dδg. Sometimes, even δ value itself is difficult to get, that is why the questions about orbital assignment still exist even for simple diatomic molecules. Our assignment illustrates the value of transient absorption measurements made in the molecular frame.
For O2, disputed structures exist around 15 eV. The adapted
absorption spectrum from34is shown by the black line in Fig.5d.
Three progressions labeled as I; I0 and I00 were observed, and converge onto the Oþ2a4Π
u threshold. The Rydberg orbital
assignment of these states has been the subject of considerable debate35,36,49,50. Similarly, Fig. 5e shows a magnified portion of
Fig. 3b. Although the bound-free transition induces a parallel modulation background, we can still see that most I peaks fall onto the weak modulation part while most I0peaks fall onto the strong modulation. The black line in Fig. 5(f) shows the asym-metry signal. By assuming3Π
u and3Σu symmetries for I and I0
respectively, the calculation shown by the red line in Fig. 5f achieved the best agreement. The effective quantum number of I0
is n¼ 2:991450, indicating the corresponding transition is
1πu! 3dπg. The3Σ
u symmetry assignment for I0conflicts with
most previous studies. A recent photodissociation study of O2
around 14.6–15.2 eV also discovered some 3Σ
u features around
vibrational states ðv ¼ 4; v ¼ 5Þ of I053. However, the most
pro-minent features of I0is v ¼ 6 and v ¼ 734, which are beyond the reported spectral range of ref. 53. The I00 state has much smaller
absorption cross section and strongly overlaps with I and I0, which makes the symmetry identification difficult. In the future, combining more highly aligned molecular ensembles and a higher
XUV spectral resolution, it shall be possible to resolve these features.
In conclusion, XUV transient absorption spectroscopy was demonstrated with aligned molecules, even though the Rydberg states have long lifetimes compared to rotational timescales28,29.
The excitation step in the molecular frame is fast, and the dependence on molecular angle is not lost despite the complex modulation of the resulting free induction decay signal. We identified the symmetry of an unassigned series in N2, and found
a conflict with previous assignment of the I0series in O2. Looking
forward, further control could be obtained by adding another NIR pulse after the XUV pulse, to induce dipole coupling between the excited electronic states, which can be controlled by alignment of the molecular target54, allowing for a clearer physical
inter-pretation of the complex absorption spectrum. We demonstrated XUV transient absorption spectroscopy in the molecular frame using linear molecules due to their simplicity in alignment. More complex molecules can be aligned in three dimensions by using a pulse sequence24, permitting similar spectroscopic studies to be
applied to symmetric top molecules. Methods
Experimental setup. In the experiment, NIR laser pulses were generated by a Ti: sapphire laser system (100 Hz, 14 mJ, 800 nm, 50 fs). Approximately 1.6 mJ was sent into a differentially pumped hollow core fiber (250 μm inner diameter) filled with 0.7 bar of argon to create a 10 fs pulse centered around 800 nm. This pulse was focused into a pulsed gas jet of xenon (3 bar backing pressure, 250 μm nozzle) where high harmonic generation took place. A thin (80 ± 10 μm) monocrystalline quartz plate was placed ~2 cm before the xenon gas jet to broaden the spectrum through self-phase modulation (SPM) and to create some second harmonic of 800 nm55,56. The intensity at the quartz plate was 3 ´ 1013W=cm2, below the
damage threshold (>4 ´ 1013W=cm2) but strong enough to generate new spectral
components through SPM. The bandwidth of each harmonic increased from 0.7 to 2.3 eV, the total spectrum spanning 12–24 eV. The XUV was composed of a train of about five pulses in an attosecond pulse train. A 500-μm-diameter pinhole was used to block the residual NIR driving field and transmit most of the XUV radiation. An intensity of 2 ´ 1011W=cm2is estimated for the residual NIR beam at
the sample target based on the Stark shift (see Supplementary Note 3 and Sup-plementray Fig. 4). All the results in this article are reproducible when we replace the pinhole with a 100-nm-thick aluminum foil (transparent energy range: 15–73 eV) or a 350-nm-thick indium foil (transparent energy range: 11–17 eV).
A part (1 mJ) of the initial 50 fs NIR pulse was combined with the XUV pulse with a holey mirror. Both the XUV and the NIR pulses propagated noncollinearly and intersected at the absorption gas jet (250 μm nozzle, gas density 1018cm 3,
interaction length 0:5 mm; N2: 99:999% purity, 10 bar backing pressure, O2:
360 a b c f e d OT I I I′ I′ I′′ WT UN UN 60 40 20 12 11 0.2 0.1 0 180 A2Π u a4Πu (Mb) (Mb) Delay (ps) Delay (ps) P (arb.u.) P (arb.u.) 9 8 0.3 Cal-UN perpendicular Cal-UN parallel Exp Cal-I′ perpendicular Cal-I′ parallel Exp 0.2 0.1 0 15.8 16 16.2
Energy (eV) Energy (eV)
16.4 16.6 14.8 15 15.2 15.4 15.6 15.8 16
Fig. 5Assigning symmetry to some Rydberg series of N2and O2. a High resolution absorption spectrum of N2adapted from25, UN refers to unassigned
states. b Zoom-in of Fig.2b. c Measured and calculated asymmetry signals. d High resolution absorption spectrum of O2adapted from ref.34. eMagnification of Fig.3b. f Measured and calculated asymmetry signals.
99:999% purity, 6.5 bar backing pressure) with a crossing angle of 18 mrad. The NIR pulse at the target (3 ´ 1013W=cm2, pulse duration increased to 70 fs due to
the dispersion of transmission optics) was used to perform impulsive alignment on the sample molecules. The transmitted XUV spectrum was dispersed in a flat-field XUV spectrometer and detected by a microchannel plate detector, with a spectral resolution of 70 meV.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes used in this study are available from the corresponding author upon reasonable request.
Received: 3 June 2019; Accepted: 29 October 2019;
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Acknowledgements
The authors thank D. Crane and R. Kroeker for technical support. The authors acknowledge fruitful discussions with X. Ding, T. J. Hammond, M. S. Schuurman, and M. Spanner. This research is supported by the NSERC Discovery Grant program. In addi-tion, we acknowledge the support of the Joint Centre for Extreme Photonics.
Author contributions
P.P. and D.M.V. conceived and planned the experiment. P.P., C.M., and M.H. conducted the measurements. P.P., C.M., and D.M.V. analyzed and interpreted the data. P.B.C., A.Y. N., and D.M.V. supervised the project. All authors discussed the results and contributed to the final manuscript.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary informationis available for this paper at https://doi.org/10.1038/s41467-019-13251-w.
Correspondenceand requests for materials should be addressed to P.P. or D.M.V.
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